Catastrophe Risk Management in a Utility Maximization Model

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1 Catastrophe Risk Management in a Utility Maximization Model Borbála Szüle Corvinus University of Budapest Hungary borbala.szule@uni-corvinus.hu Climate change may be among the factors that can contribute to the changing of weather in certain geographic regions. Sometimes extreme weather conditions are related to natural catastrophe events, such as for example floods, that can influence the economy as well. In case of a natural catastrophic event the reconstruction efforts may not only occur individually in the regions affected by the catastrophe, but aid may come also from the central government. Catastrophe insurance may also contribute to the reconstruction in case of a catastrophic event, although the economic risks of catastrophes are not always managed by catastrophe insurance. In this paper certain possibilies of catastrophe risk management are modeled in a theoretical framework. In the model a natural catastrophic event may affect one of the regions of an economy, while in other regions no catastrophe can occur. According to the assumptions, in case of a catastrophe the central government may contribute to the reconstruction by imposing a tax on the regions that have not been damaged by the catastrophe. Alternatively, if certain conditions hold, a catastrophe insurance can also be bought by the region that can be affected by the natural catastrophe. These two possibilites are compared in the paper in a theoretical model, where regions are risk averse (have a concave wealth utility function) and catastrophe insurance premium is calculated based on actuarial principles. In this theoretical framework, the method with the maximum utility of the regions can be considered as the optimal catastrophe risk management possibility. 1. Introduction Climate change may affect the economy in several ways. Some scientists have argued for example, that global warming can have an effect on the frequency of extreme weather events. Analysis of management of risks associated with these changes may be interesting, since there are numerous regions in the world that are exposed to natural catastrophes, such as for example floods or earthquakes. Depending on the severity of the natural catastrophic event, not only can the number of casualties be significant, for example also infrastructure can be damaged that generally makes also the rescue and reconstruction efforts more difficult. Natural catastrophes thus usually have far-reaching consequences also in the economy. Although among others for example also the environmental effects of catastophes can be significant, this paper focuses on risk management issues in connection with economic losses. If possible, damaging effects of natural catastrophes should be restricted preventive measures like for example building high dams may prove to be useful (against risk of floods). Unfortunately, only rarely can the occurrence of natural catastrophes be totally eliminated. In case of a catastrophic event generally reconstruction is needed so that economic activity can continue. It is also important because of the economic linkages between regions: often if one of the regions has experienced a natural catastrophic event, the effect of damages can also influence

2 other regions with which the affected region has economic relationships (for example the inhabitants of the affected region can spend less on consumption as before the catastrophe). The full economic recovery of a region after a natural catastrophe can take a long time. Obviously, economic recovery after a catastrophe can be quicker with higher financial support accessible by the region that experienced the catastrophic event. Concrete solutions to financially helping a region hit by a catastrophe may differ significantly across regions, since also the features of catastrophes and characteristics of regions are not identical. However, there are basically two main methods of financing at least part of reconstruction after a catastrophe: - one of the possibilites is a catastrophe insurance solution - the other possibility is an aid financed by a central government. Of course, also a mix of these possibilites can help in the reconstruction of a region hit by a catastrophe. The question of how to find the optimal catastrophe risk management solution in a given situation, also arises. This question is not necessarily easy to answer, since usually catastrophe losses can differ significantly across regions. This study uses a theoretical framework to try to find the optimal catastrophe risk management option. In this theoretical model expected utility of regions are compared in case of catastrophe insurance and tax financed governmental aid given to a region hit by a catastrophe. This theoretical model is relatively simple compared to the complexity of concrete empirical situations, some of the aspects of a decision about catastrophe risk management options can however be demonstrated based on the results. 2. Catastrophe events and their economic effects Catastrophes can basically be grouped into two categories: natural catastrophes and manmade disasters, the following theoretical parts of this study focus on the analysis on natural catastrophes (in other parts of the study data about both types of catastrophes occures). In that case by the way risk prevention sometimes can not be fully achieved, for example in case of earthquakes even very strong building rules may not be enough to entirely protect buildings or for example infrastructure in a region. Trying to find an optimal catastrophe risk management option can thus be considered as a relevant question in case of many regions in the world. Natural catastrophes can have different causes, the main sources of insured losses in 2010 worldwide are shown in Figure 1 (losses are measured in million USD, loss values are based on property and business interruption, excluding liability and life insurance losses). Due to the randomness of the occurrence of big natural catastrophes the main sources of losses in different years are not necessarily very similar. The size of the effect of a natural catastrophe is also influenced by (among other factors) the population growth tendencies: if a region is exposed for example to earthquakes and population growth is relatively large in that region, then (parallel to the growing population) an earthquake can have more serious consequences later when also population density (and thus may be also the number of buildings) can be higher.

3 Figure 1: Insured losses in million USD in Storms Earthquakes Floods Cold, frost Droughts, bush fires, heat waves Source: Swiss Re[2011] Catastrophe insurance is not always accessible to regions potentially exposed to natural catastrophes, and even if theoretically insurance is available, for example not every individual buys it in the given region. The value of estimated total economic losses and the cost to insurers associated to these losses can differ significantly. In 2010 for example, estimated value of economic losses of natural catastrophes and man-made disasters was approximately 218 bn USD while the cost to insurers was approximately 43 bn USD (Swiss Re 2011). From these losses in 2010 the Asian region has the largest part, as it is shown in Figure 2: Figure 2: Total economic loss by region in 2010 Seas/Space 9,5% Africa 0,2% Oceania/Australia 6,0% North America 9,4% Asia 34,3% Europe 16,1% Latin America and Caribbean 24,5% Source: Swiss Re[2011] In the interpretation of catastrophe data, certain definitions also play an important role: in the analysis of data in Figure 2 for example it should be mentioned that a (catastrophic) event is included in the Swiss Re (Sigma) statistics if insured claims, total economic losses or the number of casualties exceed a certain limit, for example this limit is 86,5 million USD in terms of total economic losses (Swiss Re 2011). Of course, for example extreme weather events can also have

4 serious consequences on the economy, and with the adoption of other limits, the concrete loss numbers could differ from those shown for example on the previously analysed Figure 2. Nevertheless, information based on these limits may also be interesting: Figure 3 for example shows the ratio of total economic loss and the GDP of given regions: Figure 3: Total economic loss in 2010 as a percentage of GDP 1,10% 0,95% 0,28% 0,19% 0,13% 0,02% Latin America and Caribbean Oceania/Australia Asia Europe North America Africa Source: Swiss Re[2011] Figure 3 refers also to the fact that the relative severity of the catastrophe depends not only on the absolute value of the losses, but also on the ability (for example measured by the GDP) of a region to help to finance at least part of the reconstruction in the region hit by the catastrophe. In case of Asia for example (that had the largest part of total economic losses experienced in 2010, where economic losses has been caused by for example extraordinary rainfalls that were followed by floods, typhoons and earthquakes) the ratio of catastrophe-related economic losses relative to the GDP is not so high as for example in the Oceania / Australia region, where economic losses were caused by for example earthquakes, floods and storms (Swiss Re 2011). 3. Modeling of insurance optimality in a utility based framework Theoretically there are some methods for dealing with catastrophes: if possible, prevention (for example not building on areas exposed to flood risk) or mitigation (for example a quick reconstruction to avoid for example infections) can prove to be useful. Catastrophe insurance can also play an important role in post-disaster financing. The availability of catastrophe insurance can be even more widespread if local insurance companies can also rely on reinsurance companies that can carry a part of the losses. In case of a catastrophe insurance the insurance premium is traditionally paid in advance (before the catastrophe can occur) and if the catastrophe event happens, a given sum is paid. This inflow of money can stimulate the economy after the catastrophe by for example playing a role in the financing of reconstruction efforts. Insurance does not necessarily exist for a given risk: there are some requirements that are to be fulfilled so that insurance can be offered by private insurance companies (Banyár 1994):

5 - each individual in the group of insured is exposed to the same risk - the group of insured is homogeneous - the number of insured should be sufficiently large. In addition to this, a risk is usually considered to be insurable if the insurance event occurs randomly and independently (in case of the insured). Independence in this context means that the probability that one of the individuals in the group of insured experiences the insurance event does not affect this probability in case of an other individual in the group of insured. Based on this traditional approach insurance is best applicable in case of independent, noncorrelated risk. The law of large numbers in case of a pool of insurance policies can be interpreted so that the larger the pool of independent risks the lower the variability of the (financial) result of the insurance company. In case of a catastrophe however sometimes the opposite of this relationship can be observed: if one of the individuals in the insurance pool is damaged by the catastrophe, the probability of damages in case of other individuals in the insurance pool is relatively high. Thus, it also means that the pooling of correlated risks increases variability in case of an insurance pool. This phenomenon is usually observable in relatively small insurance pools: if the insurance pool (the number of individuals in the insurance population) is high, the risk diversification effect as a consequence of the law of large numbers can occur. Catastrophe insurance exists in some cases, thus these insurance calculation problems (as a consequence of non-correlated risks) can sometimes be managed in practice. Catastrophe insurance however is not necessarily very cheap, and the individual exposed to a catastrophe risk can decide whether to buy catastrophe insurance. This decision can be a very complex process, but theoretically it can be modeled based on evaluating expected wealth utilities. Figure 4: Insurance premium in a theoretical model utility, in case of a catastrophe utility, no catastrophe maximum insurance premium expected utility wealth Figure 4 shows how maximum insurance premium can be calculated (if insurance is offered by insurance companies) in a simple theoretical framework, where this calculation is done based on wealth utility functions. Given the wealth utility function of an individual (in the

6 homogeneous insurance pool) one can calculate the wealth with or without the occurrence of a catastrophe. By using the probability of the catastrophe event, expected utility level can be determined and the maximum insurance premium is that value that can be subtracted from the original wealth so that the resulting wealth has exactly that utility level as the expected utility calculated with the probability of the catastrophe. In actuarial calculations insurance premium is calculated as the sum of a net premium (that corresponds to the expected value of the loss in this framework) and the insurance loading (for example it should cover administrative expenses of an insurance company). In this simple theoretical model expected value of loss as a consequence of a catastrophe is the difference between the following two values: - the original wealth - the wealth belonging to the expected utility level. In case of a convex utility function (that refers to a situation when individuals are risk seeking) insurance is not offered by insurance companies, since the maximum insurance premium that the risk seeking individual were ready to pay would not even cover the expected loss. If the utility function however is concave (the case of risk averse individuals), insurance contracts are theoretically possible, since individuals are ready to pay more than the expected loss (that corresponds to the net premium). In that case the insurance company in this theoretical model can decide whether the maximum insurance premium is enough to offer insurance. Recall however that the insurance pool should also be relatively large (the number of individuals in the insurance population should usually exceed a certain limit) so that insurance premium can be calculated with a prudent actuarial method. It is worth mentioning, that in this framework only one catastrophe can occur: it can be interpreted so that the time period in the model is calibrated so that it allows for maximum one catastrophe. The cumulation of effects of more than one catastrophic event is thus not analyzed in this theoretical model. In the following part a theoretical model is introduced in which individuals are assumed to be risk averse (have a concave wealth utility function). In this model catastrophe insurance and post-catastrophe central tax are analysed as two alternative catastrophe risk management options, and conclusions are derived about the optimality of these methods based on some simple theoretical assumptions. 4. Assumptions of the theoretical model In this theoretical model optimal catastrophe risk management options for an economic entity (for example a country) with numerous geographic regions are analyzed. In case of a large economic entity (country) usually not all regions are exposed to the same type of catastrophe risk. This feature can be modeled so that only one of the regions is exposed to a catastrophe. The number of unexposed regions is denoted by N in the model. According to the assumptions only one period is considered: during this period only one catastrophe event can occur. Similar to the model mentioned in Section 2 this theoretical model does not analyze potential accumulation of

7 wealth effects arising from more catastrophes, either. It is also assumed that economic effects of only one type of catastrophe are analyzed. The wealth of individuals can have several components in practice. An important feature of wealth is liquidity. Some components of wealth are illiquid, which means that it can not be sold in the market, or sometimes illiquidity is also mentioned in connection with wealth components that theoretically can be sold, but selling can not be done immediately. In contrast to this, in case of liquid wealth components, sale of the given asset can be immediate in the markets. In the theoretical model the wealth of regions is assumed to consist of an illiquid and a liquid part (corresponding for example a house and the income). In the absence of catastrophe the total wealth (W) of the homogeneous regions is: W = H + where H denotes the illiquid wealth and the liquid part of the wealth is denoted by L. In case of a catastrophe both parts of wealth of the region exposed to the catastrophe are affected. The total damage caused by the catastrophe in the exposed region is a random variable: L ξ = d H + (L-F), with probability p 0, with probability (1-p) where d is the ratio of the damage in case of the illiquid wealth and p is the probability of the catastrophe event, and F denotes a part of liquid wealth that is not affected by the catastrophe. In practice often also those regions are affected economically by the catastrophe that were not directly damaged. In this model, this phenomenon is modeled so that liquid assets of regions are not independent. According to the assumptions, the liquid wealth of the unexposed regions is equal to the liquid wealth of the exposed region. Individuals in the theoretical model are assumed to be the regions in the economic entity (country). This assumption reflects the phenomenon that if the insurance pool consists of for example individual households, then catastrophe losses can be correlated within a region. Of course, the definition of regions can be difficult, in this model it is assumed that an adequate determination of regions is possible, based on for example geographic features and probability of a given type of catastrophe. If the insurance pool consists of regions, correlation between losses belonging to regions may be lower (compared to the case when insurance pool consists of for example individual households), thus insurance calculations may be made more easily.

8 Figure 5: Utility in case of the region exposed to the catastrophe U(H+L-d H-(L-F)) U(H+L) p (d H-(L-F)) expected utility wealth Regions are assumed to be risk averse, thus utility of wealth of the regions is measured by a concave wealth utility function. Figure 5 illustrates the expected value of the possible loss as a consequence of a catastrophe, this expected value is p (d H + (L-F)). The concave utility function of the regions is denoted by U(W). Mathematically, in case of a concave utility function 2 du ( W ) du ( W ) the derivatives of the functions have given signs: > 0 and < 0. A possible 2 dw dw function form for the utility function is the logarithmic one: in the following U ( W ) = ln( W ) is assumed in the model. 5. Catastrophe risk management options According to the assumptions, a catastrophe event can affect the wealth of both exposed and unexposed regions. There is a wide range of possible solutions how to manage the wealth effects of a catastrophe in practice, in this theoretical model two possible catastrophe risk management options are compared: - if catastrophe insurance is available on the insurance market, the exposed region could possibly buy a catastrophe insurance that could cover the total damage - in the absence of catastrophe insurance a tax could be imposed on the regions not hit by the catastrophe event to cover a part of the total damage. An important difference between these options is that insurance premium is paid in advance (at the beginning of the period), while tax is paid at the end of the period in the model. According to the assumptions, calculation of insurance premium is based on actuarial principles, the total insurance premium is equal to the sum of the expected value of losses and the insurance loading: where c refers to the insurance loading. ( d H + L F ) p ( 1 + c)

9 If the region that is exposed to the catastrophe buys a catastrophe insurance (by assuming that this insurance is offered and the liquid wealth of the region is enough to pay the insurance premium), then the utility of this region is: ( L + H ( d H + L F ) p ( 1+ c) ) ln. In this case the utility of the regions that are not exposed to the catastrophe event is: ln ( L + H ). The other possible catastrophe risk management option plays a role in case of a catastrophe event. In the model it is assumed that a central government of the economic entity (country) can impose a tax on the regions where no catastrophe event occured so that an aid can be paid to the region hit by the catastrophe. The total transfer financed by the tax is part (x) of the damage of the illiquid assets in the model. This assumption of the model reflects the phenomenon that sometimes central government supports reconstruction of buildings and infrastructure in a region hit by a natural catastrophe. According to the assumptions, the regions where no catastrophe event occurs, pay the equal amount of tax: x d H N Given the ratio x the utility level of the region exposed to the catastrophe depends on the random variable ξ (the value of the damage), thus the expected utility level is compared with the utility in case of a catastrophe insurance. According to the assumptions in the model, the expected utility in case of a reconstruction tax : ( F + H d H + d H x) + ( 1 p) ln( L H ) p ln + The utility level of regions where no catastrophe event occurs also depends on the damage caused by the catastrophe. The expected utility level of these regions in case of a tax which is used for partly reconstruction of illiquid assets damaged by the catastrophe: d H x p ln F + H + + N ( 1 p) ln( L H ) In the theoretical model it is assumed that expected utility levels can be compared (a higher utility level can be considered as better) and in addition to this utility levels of regions can be aggregated. It is assumed that the total utility level of the economic entity (country) can be calculated as the sum of utility levels of the regions. Optimal catastrophe risk management option in this theoretical model can be identified as the option with the highest total utility level.

10 6. Optimal catastrophe risk management In case of an increase in the reconstruction tax the expected utility levels of the regions change. This tax is only imposed after a catastrophe has hit a region. 0,096 Figure 6: Utility levels of regions as a function of the rate x 0,095 0,095 0,094 0,094 0,093 0,093 0,092 0,092 0, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 value of x exposed region one of the unexposed regions Source: own calculations The utility level of the region exposed to the catastrophe and the utility level of the other regions change oppositely. Figure 6 illustrates this phenomenon (F=0.01, L=0.1, H=1, p=0.005, d=0.5). Optimum value of the reconstruction tax can be calculated based on the total utility of all regions. With the assumption that total utility of the regions can be calculated as the sum of the individual utility levels the optimal level of the tax is the value that maximizes the following expression (the sum of utility values for all groups): p ln ( F + H d H + d H x) + ( 1 p) ln( L + H ) d H x + N p ln F + H + N + ( 1 p) ln( L + H ) Maximum is calculated by calculating the first derivative of this expression, in case of an optimum it should be equal to zero. The optimum value of the reconstruction tax is: x * 1 = 1 1+ N = N N + 1 This is a simple expression and relatively easy to interpret: in this simple model framework the optimal contribution rate to the reconstruction of damages caused by a catastrophe event approaches 1 as the number of regions increases. This result thus means that the higher the number of regions not exposed to the catastrophe (compared to the number of regions hit by the

11 catastrophe, in this model there is only one such region), the larger the optimal tax-financed reconstruction support. If the optimal reconstruction tax is imposed, total utility level of regions in case of an imposed reconstruction tax is maximal. The optimality of a given catastrophe management option can also be analyzed, since for given parameters the option with the higher total utility level can be found. In the following a situation is analysed when the economic entity consists of two regions, and one of the regions is exposed to a natural catastrophe. Figure 7 illustrates a situation when total utility level in case of a reconstruction tax changes with the value of x (F=0.9, L=0.95, H=1, p=0.001, d=0.5, c=0.2, N=1). It can be observed on Figure 7 that total utility level in case of catastrophe insurance is constant, since the value of x has no effect on utility in the absence of this type of tax. The (optimal) value of x is equal to 0.5 where total utility in case if a reconstruction tax is maximal, since in this case * 1 1 x = =. Figure 7 shows a situation where the optimal reconstruction tax results in a N higher aggregate utility level than the catastrophe insurance. The parameters on Figure 7 are not necessarily representative for practical experience, the results illustrated on Figure 7 however indicate that theoretically there can be situations where total utility of the regions can be higher with reconstruction tax than with a catastrophe insurance. Figure 7: Aggregate utility level in case of different catastrophe risk management options insurance tax 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 the value of x Source: own calculations Parameter values in case of Figure 7 indicate that this situation is characterized by relatively similar values of liquid and illiquid assets, a low number of regions, and among others a relatively costly insurance). The relation of the two catastrophe risk management options changes if for example catastrophe insurance does not cost so much as in case of the parameters of Figure 7. Figure 8 illustrates a situation with the same parameters as those in case of Figure 7 except that the insurance loading is lower (F=0.9, L=0.95, H=1, p=0.001, d=0.5, c=0.05, N=1):

12 Figure 8: Aggregate utility levels with a low cost catastrophe insurance insurance tax 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 the value of x Source: own calculations With lower insurance loading (illustrated on Figure 8) the better alternative is the catastrophe insurance. These results of the theoretical model indicate that cost of catastrophe insurance is an important factor that influences the optimality of catastrophe risk management options. The results of the theoretical model were calculated based on the assumption that there is a theoretical choice between catastrophe insurance and a post-catastrophe reconstruction tax imposed on regions not hit by the natural catastrophe. Of course, this choice does not necessarily exist, not even theoretically. If an insurance company can not build a large enough (and appropriate) insurance pool, then usually no catastrophe insurance is available. In this case, in this theoretical model only a centrally imposed tax is available for financing a post-catastrophe reconstruction. In the theoretical model introduced in this paper the optimal solution (that maximizes total expected utility) can be found if for given parameters the total expected utility level is calculated for both the catastrophe insurance alternative and the reconstruction tax alternative. These alternatives differ significantly, for example in case of a catastrophe insurance the region exposed to a catastrophe pays insurance premium before a catastrophe can occur, while in case of a reconstruction tax the other (not damaged) regions pay after a catastrophe event has happened. The optimality of these two catastrophe risk management options in the theoretical model depends on the values of the model parameters. In addition to this, if catastrophe insurance theoretically proves to be better than the other alternative, then it is also necessary to analyze whether a catastrophe insurace is theoretically available. In case of an insurance the pooling of individual risks is of central importance, thus for example reinsurance companies can contribute to the availability of catastrophe insurance. 7. Conclusions Natural catastrophes can cause large economic losses, the management of catastrophe risk can thus contribute to the financial stability of the economy. The range of solutions is wide in practice, but catastrophe risk management is sometimes a mix of catastrophe insurance and government participation in the financing of reconstruction after a catastrophe. These two alternatives are compared in a theoretical model in this paper in a model framework where the

13 alternative with the higher total expected utility level is considered to be the optimal option. Based on the results of the theoretical model one of the conclusions is that the optimal rate of the reconstruction tax increases as the number of regions not affected by the catastrophe increases relative to the number of regions hit by the catastrophe. An other interesting result of the theoretical model is that the optimality of these two alternatives depend heavily on the value of the insurance loading: with higher insurance loading reconstruction tax tends to result in a higher aggregate expected utility than catastrophe insurance. With low costs in addition to the net premium (a lower insurance loading) however catastrophe insurance (if it is available) can be the optimal catastrophe risk management option (that results in a higher expected utility level). References Banyár, J.(1994): Az életbiztosítás alapjai ( Basics of life insurance, in Hungarian) Bankárképző Biztosítási Oktatási Intézet, Budapest Swiss Re (2011): Natural catastrophes and man-made disasters in 2010: a year of devastating and costly events (authors: L.Bevere, B.Rogers, B.Grollimund) Swiss Re, Sigma No. 1/2011.